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Questions tagged [linear-algebra]

For questions on linear algebra, including vector spaces, linear transformations, systems of linear equations, spanning sets, bases, dimensions and vector subspaces.

Linear algebra is concerned with vector spaces and linear transformations between them:

$$(x_1, \dots, x_n)\to a_1x_n+\dots+a_nx_n$$

Concepts include systems of linear equations, bases, dimensions, subspaces, matrices, determinants, kernels, null spaces, column spaces, traces, eigenvalues and eigenvectors, diagonalization and Jordan normal forms.

This is a general tag, most of the subjects included have secondary tags, e.g.

127034 questions
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Eigenpolynomial of a linear operator

Let $V$ be a $n$-dimensional vector space over a field $F$, let $A\in \text{End}(V)$. Let $q\in F[x]$ be an irreducible polynomial and $$ V_q:=\{x∈V:q(A)x=0\}. $$ I wish to prove that $V_q$ is not trivial iff $q$ divides the characteristic…
Richard
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how to solve a system with more equations than unkowns?

In general, how do you solve a system with more equations than unknowns? I know that if I select the equations to match them with the number of unknowns, there may be zero or many solutions depending on our selection. Where can I go from there? And…
cody
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Why doesn't this example of basis change work?

I'm learning about the change of basis in linear algebra, and trying to come up with an example to understand it. But somehow my example below doesn't make sense. Let $B_1 = ((1,0),(0,1))$ be the standard basis in $R^2$, and $v = (2,3)$ be a…
user533068
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Pairwise negative dot product implies linear independence

Let $v_1, ..., v_{m+1} \in \mathbb{R}^n$ such that $v_i \cdot v_j < 0$ if $i \ne j$. Show that $v_1, ..., v_m$ are linearly independent. My attempt: Assume $v_1, ..., v_m \in \mathbb{R}^n$ are linearly dependent with pairwise negative dot product.…
Nirusur K.
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The length of every linearly independent list of vectors is less than or equal to the length of every spanning list of vectors.

I am referring to Theorem 2.23 of the book is Linear Algebra Done Right by Axler. It mentions Theorem: In a finite-dimensional vector space, the length of every linearly independent list of vectors is less than or equal to the length of every…
TryingHardToBecomeAGoodPrSlvr
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What's the relationship between free variables and nullspaces?

I know what a free variable and a basic variable is. I understand free variables show up because there is a lack of a pivot. I understand a pivot represents a solution in a subspace that's not a nullspace. So my question is do free variables…
shawnru
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Visualizing the four subspaces of a matrix

Given a system of linear equations in the form $$AX=b$$ How can I go about visualizing the four fundamental sub-spaces - column space, row space, null space and left null space? In the same context, how can I visualize the orthogonality of row space…
Chethan Ravindranath
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Determinant of 9 Consecutive Integers

Let a $3 \times 3$ matrix have the elements $1,2,\dots,9$. What is the maximum value the determinant may have? I have found the desired value and an intuitive feeling/approach as to why that must be optimal. I struggle to really prove that claim,…
Qi Zhu
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for a $3 \times 3$ matrix A ,value of $ A^{50} $ is

I f $$A= \begin{pmatrix}1& 0 & 0 \\ 1 & 0 & 1\\ 0 & 1 & 0 \end{pmatrix}$$ then $ A^{50} $ is $$ \begin{pmatrix}1& 0 & 0 \\ 50 & 1 & 0\\ 50 & 0 & 1 \end{pmatrix}$$ $$\begin{pmatrix}1& 0 & 0 \\ 48 & 1 & 0\\ 48 & 0 & 1…
user45799
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Suppose $V$ is finite-dimensional and $E$ is a subspace of $\mathscr L(V)$

Suppose $V$ is finite-dimensional and $E$ is a subspace of $\mathscr L(V)$ such that $ST\in E$ and $TS \in E$ for all $S \in \mathscr L(V)$ and all $T\in E$. Prove that $E = \{0\}$ or $E=\mathscr L(V)$. I have started the proof, but I get lost and…
mmm
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If $A$ is $2\times2 $ matrix such that $\operatorname{tr} A =\det A=3$ then trace of $A^{-1}=$

If $A$ is $2\times2 $ matrix such that $\operatorname{tr} A=\det A=3$ then trace of $A^{-1}$ is? $(A) \quad 1 \qquad (B) \quad \dfrac{1}{3} \qquad (C) \quad \dfrac{1}{6} \qquad (D) \quad\dfrac{1}{2}$ I did it in this way: $$\lambda_1+\lambda_2…
Daman
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Properties of generalized eigenvalue problem when hermitian

This Wikipedia page says that, for the generalized eigenvalue problem $$\boldsymbol{A}\boldsymbol{v}=\lambda\boldsymbol{B}\boldsymbol{v},$$ if $\boldsymbol{A}$ and $\boldsymbol{B}$ are hermitian and $\boldsymbol{B}$ is positive-definite, then (1)…
WuQ
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how to find out a matrix for a given minimal polynomial

I know how to find out the the minimal polynomial for a given matrix. But I am stuck to do the reverse process. For example how to find out a $3\times3$ matrix, whose minimal polynomial is $x^2$.
abhi
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Finding a basis for the plane with the equation

I know the conditions of being a basis. The vectors in set should be linearly independent and they should span the vector space. So while finding a basis for the equation $y = z$, it's easy to see that the $x$ is free variable and if we call it as…
Yigit Can
  • 433
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Singular value proofs

1.) Let $A$ be a nonsingular square matrix. a.) Prove that the product of the singular values of $A$ equals the absolute value of its determinant: $\sigma_1\sigma_2...\sigma_n=|detA|$. b.) Does the sum equal the absolute value of the trace?…
diimension
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